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            <h1>情報が不完全なゲームにおける後悔最小化 (CFR)</h1>
<p>論文「<a href="http://martin.zinkevich.org/publications/regretpoker.pdf">情報が不完全なゲームにおける後悔の最小化」では、反事実に基づく後悔と、セルフプレイを通じて反事実に基づく後悔を最小限に抑えることでナッシュ均衡を実現する方法を紹介しています</a>。</p><strong>このアルゴリズムは、反事実に基づく後悔最小化 (CFR) と呼ばれています。</strong>
<p>論文「<a href="http://mlanctot.info/files/papers/nips09mccfr.pdf">大規模ゲームにおける後悔最小化のためのモンテカルロサンプリング」では、ゲームツリーからサンプリングして後悔を推定するモンテカルロ反事実的後悔最小化</a>（<strong>MCCFR</strong>）を紹介しています。</p>
<p>Pythonの実装はチュートリアルのようにわかりやすいものにしようとしました。<a href="kuhn/index.html">私たちはそれをクーンポーカーという非常に単純で不完全な情報ゲームで運営しています</a></p>。
<p><a href="https://colab.research.google.com/github/labmlai/annotated_deep_learning_paper_implementations/blob/master/labml_nn/cfr/kuhn/experiment.ipynb"><img alt="Open In Colab" src="https://colab.research.google.com/assets/colab-badge.svg"></a></p>
<p><a href="https://twitter.com/labmlai/status/1407186002255380484"><img alt="Twitter thread" src="https://img.shields.io/twitter/url?style=social&url=https%3A%2F%2Ftwitter.com%2Flabmlai%2Fstatus%2F1407186002255380484"></a>ツイッタースレッド</p>
<h2>はじめに</h2>
<p>モンテカルロ反事実後悔最小化（MCCFR）とチャンスサンプリング（CS）を実装しています。プレイヤーのすべてのアクションを試しながら、偶然の出来事をサンプリングすることで、ゲームツリーの一部を探索する反復処理を行います。チャンスイベントはディーリングカードのようなもので、イテレーションごとに1回サンプリングされます。次に、各アクションについて、<em>そのアクションを実行する代わりに現在の戦略に従った場合の後悔を計算します</em>。次に、後悔マッチングを使用して、これらの後悔に基づいて戦略を更新し、次のイテレーションに備えます。最後に、反復全体の戦略の平均を計算します。これは、十分な反復を実行した場合、ナッシュ均衡に非常に近い値になります</p>。
<p>最初に数学的表記法と理論を紹介します。</p>
<h3>プレーヤー</h3>
<p>プレーヤーはで表されます。ここで<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdn" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqen" style="">i</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeg" style="margin-right:0.10903em">N</span></span></span></span></span></span></span>、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqeg" style=""><span class="mord mathnormal" style="margin-right:0.10903em">N</span></span></span></span></span></span>はプレーヤーの集合です。</p>
<h3><a href="#History">歴史</a></h3>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdk" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqee" style=""><span class="mord mathnormal" style="margin-right:0.08125em">H</span></span></span></span></span></span>履歴は偶然の出来事を含む一連の行動であり、すべての履歴の集合です。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.13597em;"></span><span class="mord coloredeq eqej" style=""><span class="mord mathnormal" style="margin-right:0.07153em">Z</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqee" style=""><span class="mord mathnormal" style="margin-right:0.08125em">H</span></span></span></span></span></span>端末履歴 (ゲームオーバー) のセットです。</p>
<h3>アクション</h3>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span></span></span></span></span>アクション。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqce" style=""><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">:</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdk" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span><a href="#History">は端末以外の履歴です</a>。</p>
<h3><a href="#InfoSet">情報セット <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></a></h3>
<p><strong><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqct" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcw" style=""><span class="mord mathcal coloredeq eqef" style="margin-right:0.07382em">I</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdk" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span>プレーヤーに設定される情報は履歴に似ていますが</strong>、プレーヤーに表示されるアクションのみが含まれます<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>。つまり、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>履歴には相手プレイヤーに配られたカードなどのアクション/イベントが含まれますが、</p>それらはありません。
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqct" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcw" style=""><span class="mord mathcal coloredeq eqef" style="margin-right:0.07382em">I</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><strong>プレイヤーの情報パーティションとして知られています</strong><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdl" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span></span></span></span></span></span>は、特定の情報セットに属するすべての履歴の集合です。つまり、これらの履歴はすべてプレイヤーの目には同じように見えます。</p>
<p><a id="Strategy"></a></p>
<h3>ストラテジー</h3>
<p><strong><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdg" style=""><span class="mord" style=""><span class="mord" style="">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>プレイヤーのストラテジーとは</strong><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>、アクションを分散させたものです。ここで<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdo" style=""><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose" style="">)</span></span></span></span></span></span>、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdg" style=""><span class="mord" style=""><span class="mord" style="">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>はプレイヤーのすべてのストラテジーのセットです。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord coloredeq eqep" style=""><span class="mord mathnormal" style="">t</span></span></span></span></span></span>-回目の反復のストラテジーはで表されます</p>。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.943556em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqda" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>
<p>ストラテジーとは、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>与えられた情報に対して何らかのアクションを起こす確率として定義されます。</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span></span></span></span><strong>すべてのプレイヤーの戦略で構成されるストラテジープロファイルです</strong> <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span></span></span></span></span></p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord coloredeq eqcx" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span>を除くすべてのプレイヤーの戦略です <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><a id="HistoryProbability"></a></p>
<h3>歴史の確率</h3>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcu" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span></span></span></span>はストラテジープロファイルでヒストリーに到達する確率です。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.108031em;vertical-align:-0.358031em;"></span><span class="mord coloredeq eqcm" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen coloredeq eqcu" style="">(</span><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqcu" style="">)</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161964em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.358031em;"><span></span></span></span></span></span></span></span></span></span></span></span>は、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>プレイヤーの貢献なしにリーチする確率です。つまり、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span></span></span></span></span>プレイヤーが次のアクションを実行した確率です。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0497em;vertical-align:-0.29969999999999997em;"></span><span class="mord coloredeq eqco" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen coloredeq eqcu" style="">(</span><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqcu" style="">)</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161964em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29969999999999997em;"><span></span></span></span></span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>プレイヤーの貢献度だけでリーチする確率です。つまり、<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcu" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.108031em;vertical-align:-0.358031em;"></span><span class="mord coloredeq eqco" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen coloredeq eqcu" style="">(</span><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqcu" style="">)</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161964em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29969999999999997em;"><span></span></span></span></span></span></span></span><span class="mord coloredeq eqcm" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen coloredeq eqcu" style="">(</span><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqcu" style="">)</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161964em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.358031em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>情報セットに到達する確率は、<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.3794880000000003em;vertical-align:-1.329483em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdl" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.329483em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqcu" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
<h3>ユーティリティ (ペイオフ)</h3>
<p><a href="#terminal_utility">ターミナルユーティリティは</a>、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>プレイヤーのターミナル履歴に対する効用（または報酬）<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span>です。</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdq" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span>どこ <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdm" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span></span></span></span></span></span></p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcy" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="mclose" style="">)</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>ストラテジープロファイルを持つプレイヤーに期待される効用（ペイオフ）です。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span></span></span></span></p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcy" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.3794880000000003em;vertical-align:-1.329483em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdm" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.329483em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqdq" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span><span class="mord coloredeq eqcu" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
<p><a id="NashEquilibrium"></a></p>
<h3>ナッシュ・エクイリブリアム</h3>
<p>ナッシュ均衡とは、どのプレイヤーも戦略を変えるだけで期待される効用（または見返り）を上げることができない状態のことです。</p>
<p><a href="#Strategy">2人のプレイヤーにとって、ナッシュ均衡は次のような戦略プロファイルです。</a></p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0648000000000004em;vertical-align:-1.2824000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7824000000000002em;"><span style="top:-3.9424em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2824000000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7824000000000002em;"><span style="top:-3.9424em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.52566em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7416285714285715em;"><span style="top:-2.188485714285714em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3924000000000001em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.52566em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7416285714285715em;"><span style="top:-2.188485714285714em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3924000000000001em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2824000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span></span></span>-ナッシュ均衡は、</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0648000000000004em;vertical-align:-1.2824000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7824000000000002em;"><span style="top:-3.9424em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2824000000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7824000000000002em;"><span style="top:-3.9424em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.52566em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7416285714285715em;"><span style="top:-2.188485714285714em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3924000000000001em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.52566em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7416285714285715em;"><span style="top:-2.188485714285714em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3924000000000001em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2824000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><h3>後悔の最小化</h3>
<p>後悔とは、プレイヤーが最適な戦略に従わなかったり、最善の行動をとらなかったために得られなかった効用（または報酬）です。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>プレイヤーの全体的な後悔の平均は、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqei" style=""><span class="mord mathnormal" style="margin-right:0.13889em">T</span></span></span></span></span></span>すべてのラウンドで最適な戦略に従わなかったことに対する平均的な後悔です。</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.138331em;vertical-align:-0.247em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqcz" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.177714285714286em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3222857142857143em;"><span></span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight coloredeq eqdg" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3756em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord coloredeq eqcb" style=""><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqcz" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="delimsizing size2">)</span></span></span></span></span></span></span></p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord coloredeq eqep" style=""><span class="mord mathnormal" style="">t</span></span></span></span></span></span>イテレーション中の全プレイヤーのストラテジープロファイルと <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.7935559999999999em;vertical-align:0em;"></span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.093556em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcb" style=""><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqcz" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.7935559999999999em;vertical-align:0em;"></span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.94736em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqcz" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-2.441336em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span></span></span>プレイヤーのストラテジーがに置き換えられたストラテジープロファイルです。</p>
<p>平均戦略とは、すべてのラウンドで実施した戦略の平均です <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqcw" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqef" style="margin-right:0.07382em">I</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span></span></span></span></span></p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.841882em;vertical-align:-1.170941em;"></span><span class="mord coloredeq eqn" style=""><span class="mord" style=""><span class="mord" style="color:cyan"><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.670941em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mop" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8525400000000001em;"><span style="top:-2.4231360000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7253428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mop coloredeq eqbd" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqbd" style="">(</span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqbd" style="">)</span><span class="mord coloredeq eqbd" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.170941em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></span></p>
<p>それこそが、最適な戦略でプレイしなかったことに対するあからさまな後悔です。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span></span></span>すべてのプレイヤーに当てはまるなら、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqcf" style=""><span class="mord" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span> <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span></span></span>-Nash均衡です。</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:8.342229em;vertical-align:-3.9211145000000007em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.4211145em;"><span style="top:-7.3581195em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-4.8697835em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-1.4743344999999994em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9211145000000007em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.4211145em;"><span style="top:-7.3581195em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span><span style="top:-4.8697835em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqcz" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.177714285714286em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3222857142857143em;"><span></span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight coloredeq eqdg" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3756em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord coloredeq eqcb" style=""><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqcz" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="delimsizing size2">)</span></span></span></span><span style="top:-1.4743344999999994em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqcz" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.177714285714286em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3222857142857143em;"><span></span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight coloredeq eqdg" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3756em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord coloredeq eqcb" style=""><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqcz" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9211145000000007em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>ゼロサムゲームなので、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0894389999999998em;vertical-align:-0.24810799999999997em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.4518920000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span></span></span>とを加算すると、2 番目のタームがキャンセルされます。</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.395449em;vertical-align:-1.4477245em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9477245em;"><span style="top:-3.9477245em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4477245em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9477245em;"><span style="top:-3.9477245em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4477245em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>一連の戦略における効用の平均は、平均的な戦略の効用と等しくなります。</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>したがって、</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.5593910000000002em;vertical-align:-0.5296955em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0296955000000003em;"><span style="top:-3.1383645000000002em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5296955em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0296955000000003em;"><span style="top:-3.1383645000000002em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5296955em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>の定義から<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span></span></span></span></span>、<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.259391em;vertical-align:-0.36806em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>次に、</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.1187820000000004em;vertical-align:-1.3093910000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8093910000000002em;"><span style="top:-3.91806em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span><span style="top:-2.358669em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3093910000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8093910000000002em;"><span style="top:-3.91806em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.358669em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3093910000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>これが <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span></span></span>-ナッシュ均衡です。同じように、2人以上でプレイするゲームでも証明できます。</p>
<p>そのため、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span></span></span>最小化してナッシュ均衡に近づける必要があります。</p>
<p><a id="CounterfactualRegret"></a></p>
<h3>反事実に基づく後悔</h3>
<p><strong><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord" style="color:pink"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.03588em">σ</span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span></span></span></span>カウンターファクチュアルバリューは</strong>、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>プレイヤーがリーチしようとした場合<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>（<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>確率でそれに繋がるアクションを取った）プレイヤーにとって期待される効用です。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span></span></span></span></span></p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord" style="color:pink"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.03588em">σ</span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.444646em;vertical-align:-1.394641em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqeq" style=""><span class="mord mathnormal mtight" style="margin-right:0.04398em">z</span></span><span class="mrel mtight">∈</span><span class="mord mtight coloredeq eqea" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.394641em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714392em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqcx" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32808571428571426em;"><span style="top:-2.357em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.20252142857142857em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdx" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen" style="">[</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">]</span></span><span class="mclose">)</span><span class="mord coloredeq eqcj" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen coloredeq eqdx" style="">[</span><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqdx" style="">]</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqeq" style=""><span class="mord mathnormal" style="margin-right:0.04398em">z</span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>where <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqea" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> はからアクセス可能な端末履歴のセットで<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>、は up to <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdx" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen" style="">[</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">]</span></span></span></span></span></span> のプレフィックスです。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqeq" style=""><span class="mord mathnormal" style="margin-right:0.04398em">z</span></span></span></span></span></span> <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcj" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen coloredeq eqdx" style="">[</span><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqdx" style="">]</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span>は z <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdx" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen" style="">[</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">]</span></span></span></span></span></span> に到達する確率です。</p>
<p><strong>即時の反事実的後悔は</strong>、</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.274439em;vertical-align:-0.383108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.274439em;vertical-align:-0.383108em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqek" style=""><span class="mord mathnormal mtight" style="">a</span></span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight">A</span><span class="mord mtight"><span class="mord mtight coloredeq eqef" style=""><span class="mord mathnormal mtight" style="margin-right:0.07847em">I</span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.17737em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>どこ</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.274439em;vertical-align:-0.383108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord" style="color:pink"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord" style="color:pink"><span class="mord" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mord"><span class="delimsizing size2">)</span></span></span></span></span></span></span></p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqef" style=""><span class="mord mathnormal mtight" style="margin-right:0.07847em">I</span></span><span class="mrel mtight">→</span><span class="mord mtight coloredeq eqek" style=""><span class="mord mathnormal mtight" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>情報セットで常にアクションを取るように修正したストラテジープロファイルがどこにあるか</p>
<p><a href="http://martin.zinkevich.org/publications/regretpoker.pdf">論文はそれを証明しています</a>（定理3）、</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.138331em;vertical-align:-0.247em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.3717110000000003em;vertical-align:-1.321706em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqef" style=""><span class="mord mathnormal mtight" style="margin-right:0.07847em">I</span></span><span class="mrel mtight">∈</span><span class="mord mtight coloredeq eqcw" style=""><span class="mord mtight" style=""><span class="mord mathcal mtight coloredeq eqef" style="margin-right:0.07382em">I</span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.321706em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight">,</span><span class="mord mtight">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.412972em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span></span></span></span></span></span>どこ <span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.372211em;vertical-align:-0.412972em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight">,</span><span class="mord mtight">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.412972em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.274439em;vertical-align:-0.383108em;"></span><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span></span></p>
<p><a id="RegretMatching"></a></p>
<h3>リグレットマッチング</h3>
<p>ストラテジーはリグレットマッチングを使用して計算されます。</p>
<p>それぞれの情報セット、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>アクションペアに対する後悔は保たれ、</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:4.899005000000001em;vertical-align:-2.1995025000000004em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6995025000000004em;"><span style="top:-5.6842825em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord" style="color:coral"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span><span style="top:-3.1959465em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1995025000000004em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6995025000000004em;"><span style="top:-5.6842825em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord" style="color:pink"><span class="mord" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span></span></span><span style="top:-3.1959465em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1995025000000004em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>そして戦略は後悔マッチングで計算されます</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.9000399999999997em;vertical-align:-1.70002em;"></span><span class="mord coloredeq eqb" style=""><span class="mord" style=""><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2000199999999994em;"><span style="top:-4.200019999999999em;"><span class="pstrut" style="height:4.05002em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="minner" style=""><span class="mopen" style=""><span class="delimsizing mult" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05002em;"><span style="top:-2.49999em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.49199em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.889em"><svg height="0.016em" preserveaspectratio="xMinYMin" style="width:0.889em" viewbox="0 0 889 16" width="0.889em" xmlns="http://www.w3.org/2000/svg"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-3.15001em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292009999999999em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.889em"><svg height="0.016em" preserveaspectratio="xMinYMin" style="width:0.889em" viewbox="0 0 889 16" width="0.889em" xmlns="http://www.w3.org/2000/svg"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-4.30002em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span><span class="mord" style=""><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0420485em;"><span style="top:-4.0420485em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2158449999999998em;"><span style="top:-2.509755em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mop mtight coloredeq eqbi" style=""><span class="mop op-symbol small-op mtight" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.28539000000000003em;"><span style="top:-2.2853899999999996em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.6068285714285713em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8495600000000001em;"><span style="top:-2.84956em;margin-right:0.1em;"><span class="pstrut" style="height:2.55556em;"></span><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5000100000000001em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.19516666666666668em"></span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9717785714285714em;"><span style="top:-2.177714285714286em;margin-left:-0.00773em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.9836857142857145em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.32228571428571423em;"><span></span></span></span></span></span></span><span class="mopen mtight" style="color:orange">(</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct mtight" style="color:orange">,</span><span class="mord mtight" style="color:orange"><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose mtight" style="color:orange">)</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5356em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9717785714285714em;"><span style="top:-2.177714285714286em;margin-left:-0.00773em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.9836857142857145em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3222857142857143em;"><span></span></span></span></span></span></span><span class="mopen mtight" style="color:orange">(</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct mtight" style="color:orange">,</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="mclose mtight" style="color:orange">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8402519999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct" style="">,</span></span></span><span style="top:-2.1937965em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mopen mtight coloredeq eqcl" style="">∣</span><span class="mord mathnormal mtight coloredeq eqcl" style="">A</span><span class="mopen mtight coloredeq eqcl" style="">(</span><span class="mord mtight coloredeq eqcl" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight coloredeq eqcl" style="">)∣</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct" style="">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5420485em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0420485em;"><span style="top:-4.0420485em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord text" style=""><span class="mord" style=""><span class="mord coloredeq eqen" style="">i</span></span><span class="mord" style="">f</span></span><span class="mord" style=""><span class="mord coloredeq eqbg" style=""><span class="mop coloredeq eqbi" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.22528999999999993em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.47471em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose" style="color:orange">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel coloredeq eqbg" style="">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord coloredeq eqbg" style="">0</span></span></span></span><span style="top:-2.1937965em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord text" style=""><span class="mord" style="">o</span><span class="mord" style=""><span class="mord coloredeq eqep" style="">t</span></span><span class="mord" style=""><span class="mord coloredeq eqem" style="">h</span></span><span class="mord" style="">erw</span><span class="mord" style=""><span class="mord coloredeq eqen" style="">i</span></span><span class="mord" style="">se</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5420485em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.70002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span><p>どこ <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel coloredeq eqz" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqz" style=""><span class="mop coloredeq eqdv" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">(</span></span></span><span class="mord coloredeq eqz" style=""><span class="mord" style="color:orange"><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span></span><span class="mpunct coloredeq eqz" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqz" style="">0</span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">)</span></span></span></span></span></span></span></span></p>
<p><a href="#NashEquilibrium">論文「<a href="http://martin.zinkevich.org/publications/regretpoker.pdf">情報が不完全なゲームにおける後悔の最小化</a>」という論文は、上記の方程式に従って戦略を選択した場合、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span></span></span>ナッシュ均衡に比例して小さくなり、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.383108em;vertical-align:-0.538em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.537397em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322900000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight coloredeq eqei" style="padding-left:0.833em"><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span><span style="top:-2.89229em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg height="1.08em" preserveaspectratio="xMinYMin slice" viewbox="0 0 400000 1080" width="400em" xmlns="http://www.w3.org/2000/svg"><path d="M95,702
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M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.10770999999999997em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.538em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>したがってナッシュ均衡に達することを証明しています。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span></span></span></a></p>
<p><a id="MCCFR"></a></p>
<h3>モンテカルロCFR (MCCFR)</h3>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.05222em;vertical-align:-0.258664em;"></span><span class="mord" style="color:coral"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-2.441336em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>コンピューティングでは、イテレーションのたびにゲームツリー全体を拡張する必要があります。</p>
<p><a href="http://mlanctot.info/files/papers/nips09mccfr.pdf">大規模ゲームにおける後悔最小化のためのモンテカルロサンプリングという論文は、ゲームツリーからサンプリングして後悔を推定できることを示しています</a>。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.78055em;vertical-align:-0.09722em;"></span><span class="mord mathcal">Q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>は <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqej" style=""><span class="mord mathnormal" style="margin-right:0.07153em">Z</span></span></span></span></span></span> (<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqdz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqej" style=""><span class="mord mathnormal" style="margin-right:0.07153em">Z</span></span></span></span></span></span>) のサブセットの集合で、1 回の反復で 1 <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqdz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span> つのブロックしか見ることができません。すべてのサブセットのスパン <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqej" style=""><span class="mord mathnormal" style="margin-right:0.07153em">Z</span></span></span></span></span></span> () <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.55556em;vertical-align:0em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqej" style=""><span class="mord mathnormal" style="margin-right:0.07153em">Z</span></span></span></span></span></span> の和集合です。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqec" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqdz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span>ブロックを選ぶ確率です。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdw" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqeq" style=""><span class="mord mathnormal" style="margin-right:0.04398em">z</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqec" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqde" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord coloredeq eqdz" style=""><span class="mord mathnormal" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span></span>は現在のイテレーションでピッキングする確率、つまり <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdw" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.24703em;vertical-align:-0.49703em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.17862099999999997em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqeo" style=""><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span><span class="mrel mtight">:</span><span class="mord mtight coloredeq eqde" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdz" style=""><span class="mord mathnormal mtight" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.49703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqec" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span>-の合計です。</p>
<p>次に、<strong>ブロックから反事実値をサンプリングし</strong>、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqeo" style=""><span class="mord mathnormal" style="margin-right:0.05724em">j</span></span></span></span></span></span></p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord accent" style="color:pink"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord mathnormal" style="margin-right:0.03588em;color:pink">σ</span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mord" style="color:pink">∣</span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqeo" style="margin-right:0.05724em">j</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.813096em;vertical-align:-1.491656em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqde" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdz" style=""><span class="mord mathnormal mtight" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.491656em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqdw" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714392em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqcx" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32808571428571426em;"><span style="top:-2.357em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.20252142857142857em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdx" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen" style="">[</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">]</span></span><span class="mclose">)</span><span class="mord coloredeq eqcj" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen coloredeq eqdx" style="">[</span><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqdx" style="">]</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqeq" style=""><span class="mord mathnormal" style="margin-right:0.04398em">z</span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>論文はそれを示しています</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mord mathbb">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166400000000005em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqeo" style=""><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span><span class="mrel mtight">∼</span><span class="mord mtight coloredeq eqec" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.34731999999999996em;"><span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">[</span></span><span class="mord accent" style="color:pink"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord mathnormal" style="margin-right:0.03588em;color:pink">σ</span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mord" style="color:pink">∣</span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqeo" style="margin-right:0.05724em">j</span></span><span class="mclose" style="color:pink">)</span><span class="mord"><span class="delimsizing size2">]</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord" style="color:pink"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.03588em">σ</span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
<p>簡単な証拠で。</p>
<p>そのため、ゲームツリーの一部をサンプリングして後悔を計算することができます。後悔の推定値を計算します</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.093556em;vertical-align:-0.25em;"></span><span class="mord" style="color:coral"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="color:coral">(</span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:coral">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:coral">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.093556em;vertical-align:-0.25em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord" style="color:pink"><span class="mord" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.093556em;vertical-align:-0.25em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span></span></span></span></span></span></p>
<p>そして、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>それを使って各イテレーションのストラテジーを更新し<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>、計算します。最後に、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord" style="color:cyan"><span class="mord" style=""><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span></span></span></span></span>全体の平均戦略を計算します。</p>
<p>ここでは、<a href="kuhn/index.html">クーンポーカーで CFR を試すためのクーンポーカーの実装を紹介します</a>。</p>
<p><em>さっそくコードを見ていきましょう！</em></p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">328</span><span></span><span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">NewType</span><span class="p">,</span> <span class="n">Dict</span><span class="p">,</span> <span class="n">List</span><span class="p">,</span> <span class="n">Callable</span><span class="p">,</span> <span class="n">cast</span>
<span class="lineno">329</span>
<span class="lineno">330</span><span class="kn">from</span> <span class="nn">labml</span> <span class="kn">import</span> <span class="n">monit</span><span class="p">,</span> <span class="n">tracker</span><span class="p">,</span> <span class="n">logger</span><span class="p">,</span> <span class="n">experiment</span>
<span class="lineno">331</span><span class="kn">from</span> <span class="nn">labml.configs</span> <span class="kn">import</span> <span class="n">BaseConfigs</span><span class="p">,</span> <span class="n">option</span></pre></div>
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    <div class='section' id='section-1'>
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            <div class='section-link'>
                <a href='#section-1'>#</a>
            </div>
            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdn" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqen" style="">i</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeg" style="margin-right:0.10903em">N</span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqeg" style=""><span class="mord mathnormal" style="margin-right:0.10903em">N</span></span></span></span></span></span>プレイヤーの集合がどこにいるか</p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">334</span><span class="n">Player</span> <span class="o">=</span> <span class="n">NewType</span><span class="p">(</span><span class="s1">&#39;Player&#39;</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span></pre></div>
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    <div class='section' id='section-2'>
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            <div class='section-link'>
                <a href='#section-2'>#</a>
            </div>
            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span></span></span></span></span><a href="#History">アクション <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqce" style=""><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">:</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span></span> (<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdk" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span>端末以外の履歴は表示)</a></p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">336</span><span class="n">Action</span> <span class="o">=</span> <span class="n">NewType</span><span class="p">(</span><span class="s1">&#39;Action&#39;</span><span class="p">,</span> <span class="nb">str</span><span class="p">)</span></pre></div>
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    </div>
    <div class='section' id='section-3'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-3'>#</a>
            </div>
            <p><a id="History"></a></p>
<h2>歴史</h2>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdk" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqee" style=""><span class="mord mathnormal" style="margin-right:0.08125em">H</span></span></span></span></span></span>履歴は偶然の出来事を含む一連の行動であり、すべての履歴の集合です。</p>
<p>このクラスは、ゲーム固有のロジックで拡張する必要があります。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">339</span><span class="k">class</span> <span class="nc">History</span><span class="p">:</span></pre></div>
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    </div>
    <div class='section' id='section-4'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-4'>#</a>
            </div>
            <p>端末の履歴、つまりゲームオーバーかどうか。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdm" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span></span></span></span></span></span></p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">351</span>    <span class="k">def</span> <span class="nf">is_terminal</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
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    </div>
    <div class='section' id='section-5'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-5'>#</a>
            </div>
            
        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">356</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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    </div>
    <div class='section' id='section-6'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-6'>#</a>
            </div>
            <p><a id="terminal_utility"><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span></a>端末履歴用のプレーヤーのユーティリティ。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdq" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span>どこ <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdm" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span></span></span></span></span></span></p>

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            <div class="highlight"><pre><span class="lineno">358</span>    <span class="k">def</span> <span class="nf">terminal_utility</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">Player</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">float</span><span class="p">:</span></pre></div>
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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">364</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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            <p>現在のプレーヤーを取得する (<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqeh" style=""><span class="mord mathnormal" style="margin-right:0.13889em">P</span></span></span></span></span></span>という文字で表される<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdu" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeh" style="margin-right:0.13889em">P</span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span>)。<strong>これをプレーヤー関数と呼びます</strong>。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqel" style=""><span class="mord mathnormal" style="">c</span></span></span></span></span></span>現在の出来事が偶然の出来事であることを意味する場合<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdf" style=""><span class="mord" style=""><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqeh" style="margin-right:0.13889em">P</span></span><span class="mopen coloredeq eqdu" style="">(</span><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqdu" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqel" style="">c</span></span></span></span></span></span></span>。カードを配ったり、ポーカーで一般的なカードを開いたりするようなものです。</p>

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            <div class="highlight"><pre><span class="lineno">366</span>    <span class="k">def</span> <span class="nf">player</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Player</span><span class="p">:</span></pre></div>
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    <div class='section' id='section-9'>
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                <a href='#section-9'>#</a>
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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">373</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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            <p>次のステップがチャンスステップ、つまり新しいカードを配るようなものかどうか。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdf" style=""><span class="mord" style=""><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqeh" style="margin-right:0.13889em">P</span></span><span class="mopen coloredeq eqdu" style="">(</span><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqdu" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqel" style="">c</span></span></span></span></span></span></span></p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">375</span>    <span class="k">def</span> <span class="nf">is_chance</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">bool</span><span class="p">:</span></pre></div>
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    <div class='section' id='section-11'>
        <div class='docs'>
            <div class='section-link'>
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        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">380</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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    <div class='section' id='section-12'>
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            <p>次の機会にぜひお試しください<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdf" style=""><span class="mord" style=""><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqeh" style="margin-right:0.13889em">P</span></span><span class="mopen coloredeq eqdu" style="">(</span><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqdu" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqel" style="">c</span></span></span></span></span></span></span>.</p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">382</span>    <span class="k">def</span> <span class="nf">sample_chance</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Action</span><span class="p">:</span></pre></div>
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    <div class='section' id='section-13'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-13'>#</a>
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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">386</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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    <div class='section' id='section-14'>
        <div class='docs doc-strings'>
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            <p>履歴にアクションを追加します。</p>

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            <div class="highlight"><pre><span class="lineno">388</span>    <span class="k">def</span> <span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">action</span><span class="p">:</span> <span class="n">Action</span><span class="p">):</span></pre></div>
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    <div class='section' id='section-15'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-15'>#</a>
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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">392</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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    <div class='section' id='section-16'>
        <div class='docs doc-strings'>
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            <p><a href="#InfoSet">現在のプレイヤーに設定されている情報を取得</a></p>

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            <div class="highlight"><pre><span class="lineno">394</span>    <span class="k">def</span> <span class="nf">info_set_key</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">str</span><span class="p">:</span></pre></div>
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    <div class='section' id='section-17'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-17'>#</a>
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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">398</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span></pre></div>
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    <div class='section' id='section-18'>
        <div class='docs doc-strings'>
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                <a href='#section-18'>#</a>
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            <p><a href="#InfoSet">現在のプレイヤーの情報セットを新規作成</a></p>

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            <div class="highlight"><pre><span class="lineno">400</span>    <span class="k">def</span> <span class="nf">new_info_set</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="s1">&#39;InfoSet&#39;</span><span class="p">:</span></pre></div>
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    <div class='section' id='section-19'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-19'>#</a>
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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">404</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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    <div class='section' id='section-20'>
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            <p>人間が読める表現</p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">406</span>    <span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
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    <div class='section' id='section-21'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-21'>#</a>
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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">410</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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            <p><a id="InfoSet"></a></p>
<h2>情報セット <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></h2>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">413</span><span class="k">class</span> <span class="nc">InfoSet</span><span class="p">:</span></pre></div>
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    <div class='section' id='section-23'>
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            <p>情報セットを識別するユニークなキー</p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">421</span>    <span class="n">key</span><span class="p">:</span> <span class="nb">str</span></pre></div>
        </div>
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    <div class='section' id='section-24'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-24'>#</a>
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            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>、<a href="#Strategy">プレイヤーの戦略</a> <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span></p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">423</span>    <span class="n">strategy</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="n">Action</span><span class="p">,</span> <span class="nb">float</span><span class="p">]</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-25'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-25'>#</a>
            </div>
            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdo" style=""><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose" style="">)</span></span></span></span></span></span>それぞれの行動をとらなかったことに対する完全な後悔、</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:4.899005000000001em;vertical-align:-2.1995025000000004em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6995025000000004em;"><span style="top:-5.6842825em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord" style="color:coral"><span class="mord" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span><span style="top:-3.1959465em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1995025000000004em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6995025000000004em;"><span style="top:-5.6842825em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord" style="color:pink"><span class="mord" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span></span></span><span style="top:-3.1959465em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="color:coral">(</span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:coral">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:coral">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1995025000000004em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span>戦略計算時にどうせタームが相殺されてしまうので、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqbt" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mord" style=""><span class="mord" style="color:orange"><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span></span></span></span></span></span></span>代わりにメンテナンスを行います。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span></p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">438</span>    <span class="n">regret</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="n">Action</span><span class="p">,</span> <span class="nb">float</span><span class="p">]</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-26'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-26'>#</a>
            </div>
            <p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord coloredeq eqbd" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span></span></span></span></span></span>累積戦略を維持して全体の平均戦略を計算します</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.841882em;vertical-align:-1.170941em;"></span><span class="mord coloredeq eqn" style=""><span class="mord" style=""><span class="mord" style="color:cyan"><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.670941em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mop" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8525400000000001em;"><span style="top:-2.4231360000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7253428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mop coloredeq eqbd" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqbd" style="">(</span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqbd" style="">)</span><span class="mord coloredeq eqbd" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.170941em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">445</span>    <span class="n">cumulative_strategy</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="n">Action</span><span class="p">,</span> <span class="nb">float</span><span class="p">]</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-27'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-27'>#</a>
            </div>
            <p>[初期化]</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">447</span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">str</span><span class="p">):</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-28'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-28'>#</a>
            </div>
            
        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">451</span>        <span class="bp">self</span><span class="o">.</span><span class="n">key</span> <span class="o">=</span> <span class="n">key</span>
<span class="lineno">452</span>        <span class="bp">self</span><span class="o">.</span><span class="n">regret</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">actions</span><span class="p">()}</span>
<span class="lineno">453</span>        <span class="bp">self</span><span class="o">.</span><span class="n">cumulative_strategy</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">actions</span><span class="p">()}</span>
<span class="lineno">454</span>        <span class="bp">self</span><span class="o">.</span><span class="n">calculate_strategy</span><span class="p">()</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-29'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-29'>#</a>
            </div>
            <p>アクション <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdo" style=""><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose" style="">)</span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">456</span>    <span class="k">def</span> <span class="nf">actions</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">List</span><span class="p">[</span><span class="n">Action</span><span class="p">]:</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-30'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-30'>#</a>
            </div>
            
        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">460</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-31'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-31'>#</a>
            </div>
            <p>保存したディクショナリから情報セットを読み込む</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">462</span>    <span class="nd">@staticmethod</span>
<span class="lineno">463</span>    <span class="k">def</span> <span class="nf">from_dict</span><span class="p">(</span><span class="n">data</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="s1">&#39;InfoSet&#39;</span><span class="p">:</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-32'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-32'>#</a>
            </div>
            
        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">467</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-33'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-33'>#</a>
            </div>
            <p>情報セットをディクショナリに保存する</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">469</span>    <span class="k">def</span> <span class="nf">to_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-34'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-34'>#</a>
            </div>
            
        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">473</span>        <span class="k">return</span> <span class="p">{</span>
<span class="lineno">474</span>            <span class="s1">&#39;key&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">key</span><span class="p">,</span>
<span class="lineno">475</span>            <span class="s1">&#39;regret&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">regret</span><span class="p">,</span>
<span class="lineno">476</span>            <span class="s1">&#39;average_strategy&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">cumulative_strategy</span><span class="p">,</span>
<span class="lineno">477</span>        <span class="p">}</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-35'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-35'>#</a>
            </div>
            <p>保存したディクショナリからデータを読み込む</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">479</span>    <span class="k">def</span> <span class="nf">load_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">data</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">]):</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-36'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-36'>#</a>
            </div>
            
        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">483</span>        <span class="bp">self</span><span class="o">.</span><span class="n">regret</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="s1">&#39;regret&#39;</span><span class="p">]</span>
<span class="lineno">484</span>        <span class="bp">self</span><span class="o">.</span><span class="n">cumulative_strategy</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="s1">&#39;average_strategy&#39;</span><span class="p">]</span>
<span class="lineno">485</span>        <span class="bp">self</span><span class="o">.</span><span class="n">calculate_strategy</span><span class="p">()</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-37'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-37'>#</a>
            </div>
            <h2>ストラテジーの計算</h2>
<p><a href="#RegretMatching">リグレットマッチングを使用して現在の戦略を計算します</a>。</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.9000399999999997em;vertical-align:-1.70002em;"></span><span class="mord coloredeq eqb" style=""><span class="mord" style=""><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2000199999999994em;"><span style="top:-4.200019999999999em;"><span class="pstrut" style="height:4.05002em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="minner" style=""><span class="mopen" style=""><span class="delimsizing mult" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05002em;"><span style="top:-2.49999em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.49199em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.889em"><svg height="0.016em" preserveaspectratio="xMinYMin" style="width:0.889em" viewbox="0 0 889 16" width="0.889em" xmlns="http://www.w3.org/2000/svg"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-3.15001em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292009999999999em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.889em"><svg height="0.016em" preserveaspectratio="xMinYMin" style="width:0.889em" viewbox="0 0 889 16" width="0.889em" xmlns="http://www.w3.org/2000/svg"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-4.30002em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span><span class="mord" style=""><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0420485em;"><span style="top:-4.0420485em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2158449999999998em;"><span style="top:-2.509755em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mop mtight coloredeq eqbi" style=""><span class="mop op-symbol small-op mtight" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.28539000000000003em;"><span style="top:-2.2853899999999996em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.6068285714285713em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8495600000000001em;"><span style="top:-2.84956em;margin-right:0.1em;"><span class="pstrut" style="height:2.55556em;"></span><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5000100000000001em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.19516666666666668em"></span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9717785714285714em;"><span style="top:-2.177714285714286em;margin-left:-0.00773em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.9836857142857145em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.32228571428571423em;"><span></span></span></span></span></span></span><span class="mopen mtight" style="color:orange">(</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct mtight" style="color:orange">,</span><span class="mord mtight" style="color:orange"><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose mtight" style="color:orange">)</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5356em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9717785714285714em;"><span style="top:-2.177714285714286em;margin-left:-0.00773em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.9836857142857145em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3222857142857143em;"><span></span></span></span></span></span></span><span class="mopen mtight" style="color:orange">(</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct mtight" style="color:orange">,</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="mclose mtight" style="color:orange">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8402519999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct" style="">,</span></span></span><span style="top:-2.1937965em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mopen mtight coloredeq eqcl" style="">∣</span><span class="mord mathnormal mtight coloredeq eqcl" style="">A</span><span class="mopen mtight coloredeq eqcl" style="">(</span><span class="mord mtight coloredeq eqcl" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight coloredeq eqcl" style="">)∣</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct" style="">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5420485em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0420485em;"><span style="top:-4.0420485em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord text" style=""><span class="mord" style=""><span class="mord coloredeq eqen" style="">i</span></span><span class="mord" style="">f</span></span><span class="mord" style=""><span class="mord coloredeq eqbg" style=""><span class="mop coloredeq eqbi" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.22528999999999993em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.47471em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose" style="color:orange">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel coloredeq eqbg" style="">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord coloredeq eqbg" style="">0</span></span></span></span><span style="top:-2.1937965em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord text" style=""><span class="mord" style="">o</span><span class="mord" style=""><span class="mord coloredeq eqep" style="">t</span></span><span class="mord" style=""><span class="mord coloredeq eqem" style="">h</span></span><span class="mord" style="">erw</span><span class="mord" style=""><span class="mord coloredeq eqen" style="">i</span></span><span class="mord" style="">se</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5420485em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.70002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span><p>どこ <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel coloredeq eqz" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqz" style=""><span class="mop coloredeq eqdv" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">(</span></span></span><span class="mord coloredeq eqz" style=""><span class="mord" style="color:orange"><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span></span><span class="mpunct coloredeq eqz" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqz" style="">0</span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">)</span></span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">487</span>    <span class="k">def</span> <span class="nf">calculate_strategy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-38'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-38'>#</a>
            </div>
            <p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9592389999999998em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.180908em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel coloredeq eqz" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqz" style=""><span class="mop coloredeq eqdv" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">(</span></span></span><span class="mord coloredeq eqz" style=""><span class="mord" style="color:orange"><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span></span><span class="mpunct coloredeq eqz" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqz" style="">0</span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">)</span></span></span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">506</span>        <span class="n">regret</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="nb">max</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">regret</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-39'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-39'>#</a>
            </div>
            <p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.5660100000000003em;vertical-align:-1.516005em;"></span><span class="mord"><span class="mop op-limits coloredeq eqbi" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.516005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9592389999999998em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.180908em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose" style="color:orange">)</span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">508</span>        <span class="n">regret_sum</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">regret</span><span class="o">.</span><span class="n">values</span><span class="p">())</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-40'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-40'>#</a>
            </div>
            <p>もし<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.433949em;vertical-align:-0.47471em;"></span><span class="mord coloredeq eqbg" style=""><span class="mord" style=""><span class="mop coloredeq eqbi" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.22528999999999993em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.47471em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose" style="color:orange">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style="">0</span></span></span></span></span></span>、</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">510</span>        <span class="k">if</span> <span class="n">regret_sum</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-41'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-41'>#</a>
            </div>
            <p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.141331em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.960188em;vertical-align:-1.323949em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.636239em;"><span style="top:-2.150761em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mop coloredeq eqbi" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.22528999999999993em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.47471em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6778919999999999em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose" style="color:orange">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.6769999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.323949em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">513</span>            <span class="bp">self</span><span class="o">.</span><span class="n">strategy</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="n">r</span> <span class="o">/</span> <span class="n">regret_sum</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">regret</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-42'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-42'>#</a>
            </div>
            <p>それ以外の場合は、</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">515</span>        <span class="k">else</span><span class="p">:</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-43'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-43'>#</a>
            </div>
            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcl" style=""><span class="mopen" style="">∣</span><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)∣</span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">517</span>            <span class="n">count</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">regret</span><span class="p">))</span></pre></div>
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    </div>
    <div class='section' id='section-44'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-44'>#</a>
            </div>
            <p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.141331em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqcl" style=""><span class="mopen" style="">∣</span><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)∣</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">520</span>            <span class="bp">self</span><span class="o">.</span><span class="n">strategy</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">count</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">regret</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-45'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-45'>#</a>
            </div>
            <h2>平均的な戦略を取得</h2>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.841882em;vertical-align:-1.170941em;"></span><span class="mord coloredeq eql" style=""><span class="mord" style=""><span class="mord" style="color:cyan"><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.670941em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mop" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8525400000000001em;"><span style="top:-2.4231360000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7253428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mop coloredeq eqbd" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqbd" style="">(</span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqbd" style="">)</span><span class="mord coloredeq eqbd" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.170941em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">522</span>    <span class="k">def</span> <span class="nf">get_average_strategy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-46'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-46'>#</a>
            </div>
            <p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">531</span>        <span class="n">cum_strategy</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">cumulative_strategy</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mf">0.</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">actions</span><span class="p">()}</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-47'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-47'>#</a>
            </div>
            <p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.344341em;vertical-align:-1.516005em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqek" style=""><span class="mord mathnormal mtight" style="">a</span></span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight">A</span><span class="mopen mtight">(</span><span class="mord mtight coloredeq eqef" style=""><span class="mord mathnormal mtight" style="margin-right:0.07847em">I</span></span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.516005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">535</span>        <span class="n">strategy_sum</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">cum_strategy</span><span class="o">.</span><span class="n">values</span><span class="p">())</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-48'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-48'>#</a>
            </div>
            <p>もし<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.2809409999999999em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span>、</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">537</span>        <span class="k">if</span> <span class="n">strategy_sum</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-49'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-49'>#</a>
            </div>
            <p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.841882em;vertical-align:-1.170941em;"></span><span class="mord coloredeq eql" style=""><span class="mord" style=""><span class="mord" style="color:cyan"><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.670941em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mop" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8525400000000001em;"><span style="top:-2.4231360000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7253428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mop coloredeq eqbd" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqbd" style="">(</span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqbd" style="">)</span><span class="mord coloredeq eqbd" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.170941em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">541</span>            <span class="k">return</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="n">s</span> <span class="o">/</span> <span class="n">strategy_sum</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">cum_strategy</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span></pre></div>
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    <div class='section' id='section-50'>
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            <p>それ以外の場合は、</p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">543</span>        <span class="k">else</span><span class="p">:</span></pre></div>
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            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcl" style=""><span class="mopen" style="">∣</span><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)∣</span></span></span></span></span></span></p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">545</span>            <span class="n">count</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">cum_strategy</span><span class="p">))</span></pre></div>
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            <div class='section-link'>
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            <p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mord" style="color:cyan"><span class="mord" style=""><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqcl" style=""><span class="mopen" style="">∣</span><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)∣</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></p>

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        <div class='code'>
            <div class="highlight"><pre><span class="lineno">548</span>            <span class="k">return</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">count</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">cum_strategy</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span></pre></div>
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    </div>
    <div class='section' id='section-53'>
        <div class='docs doc-strings'>
            <div class='section-link'>
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            </div>
            <p>人間が読める表現</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">550</span>    <span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
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                <a href='#section-54'>#</a>
            </div>
            
        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">554</span>        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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    <div class='section' id='section-55'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-55'>#</a>
            </div>
            <h2>反事実に基づく後悔最小化 (CFR) アルゴリズム</h2>
<p>すべてのチャンスイベント（ノード）をサンプリングし、他のすべてのイベント（ノード）を調査するチャンスサンプリング（<strong>CS</strong>）を行います。</p>
<p>この用語は無視できます。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdw" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span>なぜなら、チャンスサンプリングを行っていて、（分子と分母で共通）戦略を計算するときに相殺されてしまうので、すべての端末履歴で同じだからです。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">557</span><span class="k">class</span> <span class="nc">CFR</span><span class="p">:</span></pre></div>
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            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqcw" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqef" style="margin-right:0.07382em">I</span></span></span></span></span></span></span>すべての情報セットのセット。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">570</span>    <span class="n">info_sets</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">InfoSet</span><span class="p">]</span></pre></div>
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    </div>
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            </div>
            <ul><li><code  class="highlight"><span></span><span class="n">create_new_history</span></code>
新しい空の履歴を作成します</li>
<li><code  class="highlight"><span></span><span class="n">epochs</span></code>
はトレーニングする反復回数です <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqei" style=""><span class="mord mathnormal" style="margin-right:0.13889em">T</span></span></span></span></span></span></li>
<li><code  class="highlight"><span></span><span class="n">n_players</span></code>
はプレイヤーの数</li></ul>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">572</span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span>
<span class="lineno">573</span>                 <span class="n">create_new_history</span><span class="p">:</span> <span class="n">Callable</span><span class="p">[[],</span> <span class="n">History</span><span class="p">],</span>
<span class="lineno">574</span>                 <span class="n">epochs</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
<span class="lineno">575</span>                 <span class="n">n_players</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">2</span><span class="p">):</span></pre></div>
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        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">581</span>        <span class="bp">self</span><span class="o">.</span><span class="n">n_players</span> <span class="o">=</span> <span class="n">n_players</span>
<span class="lineno">582</span>        <span class="bp">self</span><span class="o">.</span><span class="n">epochs</span> <span class="o">=</span> <span class="n">epochs</span>
<span class="lineno">583</span>        <span class="bp">self</span><span class="o">.</span><span class="n">create_new_history</span> <span class="o">=</span> <span class="n">create_new_history</span></pre></div>
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    <div class='section' id='section-59'>
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                <a href='#section-59'>#</a>
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            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqcw" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqef" style="margin-right:0.07382em">I</span></span></span></span></span></span></span>すべての情報セットのセット用の辞書</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">585</span>        <span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span> <span class="o">=</span> <span class="p">{}</span></pre></div>
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    </div>
    <div class='section' id='section-60'>
        <div class='docs'>
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                <a href='#section-60'>#</a>
            </div>
            <p>分析用トラッカー</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">587</span>        <span class="bp">self</span><span class="o">.</span><span class="n">tracker</span> <span class="o">=</span> <span class="n">InfoSetTracker</span><span class="p">()</span></pre></div>
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    </div>
    <div class='section' id='section-61'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-61'>#</a>
            </div>
            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>特定の履歴における現在のプレーヤーの情報セットを返します <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">589</span>    <span class="k">def</span> <span class="nf">_get_info_set</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">h</span><span class="p">:</span> <span class="n">History</span><span class="p">):</span></pre></div>
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        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">593</span>        <span class="n">info_set_key</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">info_set_key</span><span class="p">()</span>
<span class="lineno">594</span>        <span class="k">if</span> <span class="n">info_set_key</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span><span class="p">:</span>
<span class="lineno">595</span>            <span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span><span class="p">[</span><span class="n">info_set_key</span><span class="p">]</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">new_info_set</span><span class="p">()</span>
<span class="lineno">596</span>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span><span class="p">[</span><span class="n">info_set_key</span><span class="p">]</span></pre></div>
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    </div>
    <div class='section' id='section-63'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-63'>#</a>
            </div>
            <h3>ウォークツリー</h3>
<p>この関数はゲームツリーを歩き回ります。</p>
<ul><li><code  class="highlight"><span></span><span class="n">h</span></code>
現在の履歴です <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span></li>
<li><code  class="highlight"><span></span><span class="n">i</span></code>
<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>後悔の気持ちを込めて計算している選手は</li>
<li><a href="#HistoryProbability"><code  class="highlight"><span></span><span class="n">pi_i</span></code>
</a>は <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.2309039999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqck" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span></li>
</ul><li><a href="#HistoryProbability"><code  class="highlight"><span></span><span class="n">pi_neg_i</span></code>
</a>は <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.289235em;vertical-align:-0.316995em;"></span><span class="mord coloredeq eqcg" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.316995em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span></li>
<p>この関数は、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.450201em;vertical-align:-1.400196em;"></span><span class="mord coloredeq eqbp" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqeb" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>プレフィックス付きの端末履歴の集合である履歴に対して、期待どおりのユーティリティを返します。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span></p>
<p>ティーを歩いていると、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>後悔の気持ちが全部アップデートされます。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">598</span>    <span class="k">def</span> <span class="nf">walk_tree</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">h</span><span class="p">:</span> <span class="n">History</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">Player</span><span class="p">,</span> <span class="n">pi_i</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span> <span class="n">pi_neg_i</span><span class="p">:</span> <span class="nb">float</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">float</span><span class="p">:</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-64'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-64'>#</a>
            </div>
            <p>端末履歴の場合は、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdm" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span></span></span></span></span></span>端末ユーティリティを返します<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdq" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span>。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">619</span>        <span class="k">if</span> <span class="n">h</span><span class="o">.</span><span class="n">is_terminal</span><span class="p">():</span>
<span class="lineno">620</span>            <span class="k">return</span> <span class="n">h</span><span class="o">.</span><span class="n">terminal_utility</span><span class="p">(</span><span class="n">i</span><span class="p">)</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-65'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-65'>#</a>
            </div>
            <p>それが偶然の出来事なら、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdf" style=""><span class="mord" style=""><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqeh" style="margin-right:0.13889em">P</span></span><span class="mopen coloredeq eqdu" style="">(</span><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqdu" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqel" style="">c</span></span></span></span></span></span></span> aをサンプルして次のステップに進んでください。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">622</span>        <span class="k">elif</span> <span class="n">h</span><span class="o">.</span><span class="n">is_chance</span><span class="p">():</span>
<span class="lineno">623</span>            <span class="n">a</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">sample_chance</span><span class="p">()</span>
<span class="lineno">624</span>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">walk_tree</span><span class="p">(</span><span class="n">h</span> <span class="o">+</span> <span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">pi_i</span><span class="p">,</span> <span class="n">pi_neg_i</span><span class="p">)</span></pre></div>
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    </div>
    <div class='section' id='section-66'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-66'>#</a>
            </div>
            <p>現在のプレイヤーの情報設定を取得 <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">627</span>        <span class="n">I</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_info_set</span><span class="p">(</span><span class="n">h</span><span class="p">)</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-67'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-67'>#</a>
            </div>
            <p>保存する <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.15557em;vertical-align:-0.40557em;"></span><span class="mord coloredeq eqbp" style=""><span class="mop" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.17862099999999992em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.40557em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">629</span>        <span class="n">v</span> <span class="o">=</span> <span class="mi">0</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-68'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-68'>#</a>
            </div>
            <p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.450201em;vertical-align:-1.400196em;"></span><span class="mord coloredeq eqbe" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord mtight" style=""><span class="mord mtight" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span>アクションごとに保存するには <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mclose">)</span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">633</span>        <span class="n">va</span> <span class="o">=</span> <span class="p">{}</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-69'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-69'>#</a>
            </div>
            <p>すべてのアクションを反復処理</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">636</span>        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">I</span><span class="o">.</span><span class="n">actions</span><span class="p">():</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-70'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-70'>#</a>
            </div>
            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>現在のプレーヤーが</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">638</span>            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">h</span><span class="o">.</span><span class="n">player</span><span class="p">():</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-71'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-71'>#</a>
            </div>
            <span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.3644800000000004em;vertical-align:-1.4322400000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9322400000000002em;"><span style="top:-3.93224em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.02224em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4322400000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9322400000000002em;"><span style="top:-3.93224em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqck" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span><span class="mord coloredeq eqda" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcg" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.02224em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4322400000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">643</span>                <span class="n">va</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">walk_tree</span><span class="p">(</span><span class="n">h</span> <span class="o">+</span> <span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">pi_i</span> <span class="o">*</span> <span class="n">I</span><span class="o">.</span><span class="n">strategy</span><span class="p">[</span><span class="n">a</span><span class="p">],</span> <span class="n">pi_neg_i</span><span class="p">)</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-72'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-72'>#</a>
            </div>
            <p>それ以外の場合は、</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">645</span>            <span class="k">else</span><span class="p">:</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-73'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-73'>#</a>
            </div>
            <span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.3644800000000004em;vertical-align:-1.4322400000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9322400000000002em;"><span style="top:-3.93224em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.02224em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4322400000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9322400000000002em;"><span style="top:-3.93224em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqck" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcg" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.02224em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqda" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4322400000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">650</span>                <span class="n">va</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">walk_tree</span><span class="p">(</span><span class="n">h</span> <span class="o">+</span> <span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">pi_i</span><span class="p">,</span> <span class="n">pi_neg_i</span> <span class="o">*</span> <span class="n">I</span><span class="o">.</span><span class="n">strategy</span><span class="p">[</span><span class="n">a</span><span class="p">])</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-74'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-74'>#</a>
            </div>
            <p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.450201em;vertical-align:-1.400196em;"></span><span class="mord coloredeq eqbp" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.266005em;vertical-align:-1.516005em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqek" style=""><span class="mord mathnormal mtight" style="">a</span></span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight">A</span><span class="mopen mtight">(</span><span class="mord mtight coloredeq eqef" style=""><span class="mord mathnormal mtight" style="margin-right:0.07847em">I</span></span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.516005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size4">[</span></span><span class="mord coloredeq eqda" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span><span class="mord coloredeq eqbe" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord mtight" style=""><span class="mord mtight" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mord"><span class="delimsizing size4">]</span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">655</span>            <span class="n">v</span> <span class="o">=</span> <span class="n">v</span> <span class="o">+</span> <span class="n">I</span><span class="o">.</span><span class="n">strategy</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">*</span> <span class="n">va</span><span class="p">[</span><span class="n">a</span><span class="p">]</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-75'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-75'>#</a>
            </div>
            <p>現在のプレイヤーがそうなら<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>、累積攻略法と総後悔を更新</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">659</span>        <span class="k">if</span> <span class="n">h</span><span class="o">.</span><span class="n">player</span><span class="p">()</span> <span class="o">==</span> <span class="n">i</span><span class="p">:</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-76'>
        <div class='docs'>
            <div class='section-link'>
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            <p>累積戦略の更新 <span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord coloredeq eqbd" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.157819em;vertical-align:-1.329483em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">[</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdl" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.329483em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mclose">)</span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span><span class="mord"><span class="delimsizing size2">]</span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">664</span>            <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">I</span><span class="o">.</span><span class="n">actions</span><span class="p">():</span>
<span class="lineno">665</span>                <span class="n">I</span><span class="o">.</span><span class="n">cumulative_strategy</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">I</span><span class="o">.</span><span class="n">cumulative_strategy</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">+</span> <span class="n">pi_i</span> <span class="o">*</span> <span class="n">I</span><span class="o">.</span><span class="n">strategy</span><span class="p">[</span><span class="n">a</span><span class="p">]</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-77'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-77'>#</a>
            </div>
            <span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:7.749200999999999em;vertical-align:-3.6246005em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1246005em;"><span style="top:-7.1093805em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord" style="color:coral"><span class="mord" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span><span style="top:-5.2993805em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"></span></span><span style="top:-1.7708485em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord coloredeq eqbt" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mord" style=""><span class="mord" style="color:orange"><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6246005em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1246005em;"><span style="top:-7.1093805em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord" style="color:pink"><span class="mord" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span></span></span><span style="top:-5.2993805em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.02224em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mclose">)</span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord coloredeq eqbe" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord mtight" style=""><span class="mord mtight" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqbp" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mord"><span class="delimsizing size2">)</span></span></span></span><span style="top:-1.7708485em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="color:coral">(</span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:coral">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:coral">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6246005em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">678</span>            <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">I</span><span class="o">.</span><span class="n">actions</span><span class="p">():</span>
<span class="lineno">679</span>                <span class="n">I</span><span class="o">.</span><span class="n">regret</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">+=</span> <span class="n">pi_neg_i</span> <span class="o">*</span> <span class="p">(</span><span class="n">va</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">-</span> <span class="n">v</span><span class="p">)</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-78'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-78'>#</a>
            </div>
            <p>ストラテジーを更新 <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.043556em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">682</span>            <span class="n">I</span><span class="o">.</span><span class="n">calculate_strategy</span><span class="p">()</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-79'>
        <div class='docs'>
            <div class='section-link'>
                <a href='#section-79'>#</a>
            </div>
            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>プレイヤーに期待通りのユーティリティを返し、<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.450201em;vertical-align:-1.400196em;"></span><span class="mord coloredeq eqbp" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">686</span>        <span class="k">return</span> <span class="n">v</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-80'>
        <div class='docs doc-strings'>
            <div class='section-link'>
                <a href='#section-80'>#</a>
            </div>
            <h3>繰り返し更新 <span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.043556em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span></h3>
<p>これにより、<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqei" style=""><span class="mord mathnormal" style="margin-right:0.13889em">T</span></span></span></span></span></span>イテレーションのストラテジーが更新されます。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">688</span>    <span class="k">def</span> <span class="nf">iterate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
        </div>
    </div>
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            <p><code  class="highlight"><span></span><span class="n">epochs</span></code>
ループ・フォー・タイムズ</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">696</span>        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">monit</span><span class="o">.</span><span class="n">iterate</span><span class="p">(</span><span class="s1">&#39;Train&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">epochs</span><span class="p">):</span></pre></div>
        </div>
    </div>
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                <a href='#section-82'>#</a>
            </div>
            <p>ツリーを歩き、各プレイヤーの悔い改めを更新</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">698</span>            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_players</span><span class="p">):</span>
<span class="lineno">699</span>                <span class="bp">self</span><span class="o">.</span><span class="n">walk_tree</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">create_new_history</span><span class="p">(),</span> <span class="n">cast</span><span class="p">(</span><span class="n">Player</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span></pre></div>
        </div>
    </div>
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            </div>
            <p>分析用トラッキングデータ</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">702</span>            <span class="n">tracker</span><span class="o">.</span><span class="n">add_global_step</span><span class="p">()</span>
<span class="lineno">703</span>            <span class="bp">self</span><span class="o">.</span><span class="n">tracker</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span><span class="p">)</span>
<span class="lineno">704</span>            <span class="n">tracker</span><span class="o">.</span><span class="n">save</span><span class="p">()</span></pre></div>
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    </div>
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        <div class='docs'>
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            </div>
            <p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">000</span></span></span></span></span>反復ごとにチェックポイントを保存</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">707</span>            <span class="k">if</span> <span class="p">(</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">1_000</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="lineno">708</span>                <span class="n">experiment</span><span class="o">.</span><span class="n">save_checkpoint</span><span class="p">()</span></pre></div>
        </div>
    </div>
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                <a href='#section-85'>#</a>
            </div>
            <p>情報セットを印刷</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">711</span>        <span class="n">logger</span><span class="o">.</span><span class="n">inspect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span><span class="p">)</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-86'>
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            </div>
            <h3>情報セットトラッカー</h3>
<p>これは、情報セットのデータを追跡するための小さなヘルパークラスです。</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">714</span><span class="k">class</span> <span class="nc">InfoSetTracker</span><span class="p">:</span></pre></div>
        </div>
    </div>
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        <div class='docs doc-strings'>
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            </div>
            <p>トラッキングインジケーターを設定</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">720</span>    <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
        </div>
    </div>
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                <a href='#section-88'>#</a>
            </div>
            
        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">724</span>        <span class="n">tracker</span><span class="o">.</span><span class="n">set_histogram</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;strategy.*&#39;</span><span class="p">)</span>
<span class="lineno">725</span>        <span class="n">tracker</span><span class="o">.</span><span class="n">set_histogram</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;average_strategy.*&#39;</span><span class="p">)</span>
<span class="lineno">726</span>        <span class="n">tracker</span><span class="o">.</span><span class="n">set_histogram</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;regret.*&#39;</span><span class="p">)</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-89'>
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            </div>
            <p>すべての情報セットのデータを追跡</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">728</span>    <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">info_sets</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">InfoSet</span><span class="p">]):</span></pre></div>
        </div>
    </div>
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                <a href='#section-90'>#</a>
            </div>
            
        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">732</span>        <span class="k">for</span> <span class="n">I</span> <span class="ow">in</span> <span class="n">info_sets</span><span class="o">.</span><span class="n">values</span><span class="p">():</span>
<span class="lineno">733</span>            <span class="n">avg_strategy</span> <span class="o">=</span> <span class="n">I</span><span class="o">.</span><span class="n">get_average_strategy</span><span class="p">()</span>
<span class="lineno">734</span>            <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">I</span><span class="o">.</span><span class="n">actions</span><span class="p">():</span>
<span class="lineno">735</span>                <span class="n">tracker</span><span class="o">.</span><span class="n">add</span><span class="p">({</span>
<span class="lineno">736</span>                    <span class="sa">f</span><span class="s1">&#39;strategy.</span><span class="si">{</span><span class="n">I</span><span class="o">.</span><span class="n">key</span><span class="si">}</span><span class="s1">.</span><span class="si">{</span><span class="n">a</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">:</span> <span class="n">I</span><span class="o">.</span><span class="n">strategy</span><span class="p">[</span><span class="n">a</span><span class="p">],</span>
<span class="lineno">737</span>                    <span class="sa">f</span><span class="s1">&#39;average_strategy.</span><span class="si">{</span><span class="n">I</span><span class="o">.</span><span class="n">key</span><span class="si">}</span><span class="s1">.</span><span class="si">{</span><span class="n">a</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">:</span> <span class="n">avg_strategy</span><span class="p">[</span><span class="n">a</span><span class="p">],</span>
<span class="lineno">738</span>                    <span class="sa">f</span><span class="s1">&#39;regret.</span><span class="si">{</span><span class="n">I</span><span class="o">.</span><span class="n">key</span><span class="si">}</span><span class="s1">.</span><span class="si">{</span><span class="n">a</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">:</span> <span class="n">I</span><span class="o">.</span><span class="n">regret</span><span class="p">[</span><span class="n">a</span><span class="p">],</span>
<span class="lineno">739</span>                <span class="p">})</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-91'>
        <div class='docs doc-strings'>
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            </div>
            <h3>設定可能な CFR モジュール</h3>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">742</span><span class="k">class</span> <span class="nc">CFRConfigs</span><span class="p">(</span><span class="n">BaseConfigs</span><span class="p">):</span></pre></div>
        </div>
    </div>
    <div class='section' id='section-92'>
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            </div>
            
        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">746</span>    <span class="n">create_new_history</span><span class="p">:</span> <span class="n">Callable</span><span class="p">[[],</span> <span class="n">History</span><span class="p">]</span>
<span class="lineno">747</span>    <span class="n">epochs</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">1_00_000</span>
<span class="lineno">748</span>    <span class="n">cfr</span><span class="p">:</span> <span class="n">CFR</span> <span class="o">=</span> <span class="s1">&#39;simple_cfr&#39;</span></pre></div>
        </div>
    </div>
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        <div class='docs doc-strings'>
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            </div>
            <p><strong>CFR</strong> アルゴリズムを初期化</p>

        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">751</span><span class="nd">@option</span><span class="p">(</span><span class="n">CFRConfigs</span><span class="o">.</span><span class="n">cfr</span><span class="p">)</span>
<span class="lineno">752</span><span class="k">def</span> <span class="nf">simple_cfr</span><span class="p">(</span><span class="n">c</span><span class="p">:</span> <span class="n">CFRConfigs</span><span class="p">):</span></pre></div>
        </div>
    </div>
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            </div>
            
        </div>
        <div class='code'>
            <div class="highlight"><pre><span class="lineno">756</span>    <span class="k">return</span> <span class="n">CFR</span><span class="p">(</span><span class="n">create_new_history</span><span class="o">=</span><span class="n">c</span><span class="o">.</span><span class="n">create_new_history</span><span class="p">,</span>
<span class="lineno">757</span>               <span class="n">epochs</span><span class="o">=</span><span class="n">c</span><span class="o">.</span><span class="n">epochs</span><span class="p">)</span></pre></div>
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